MaBloWriMo 9: omega and its ilk

So far, we have defined a sequence of numbers s_n, and showed that

s_n = \omega^{2^n} + \overline{\omega}^{2^n}

where \omega = 2 + \sqrt 3 and \overline{\omega} = 2 - \sqrt 3. This is a big step: the s_n are defined recursively (that is, each s_n is defined in terms of the previous s_{n-1}), but \omega and \overline{\omega} give us a direct arithmetic expression for s_n, which can make proving things about the s_n easier. So our next goal will be to better understand \omega.

As a warmup, consider that \omega and \overline{\omega} are both of the form a + b\sqrt 3, where a and b are integers. Consider the set of all such numbers, call it S. So, for example, S contains things like 4 - 19\sqrt{3} and 0 + \sqrt{3} and 99 + 999\sqrt{3} and 2 + 0\sqrt{3}… and so on. (Mathematicians would call this set \mathbb{Z}[\sqrt{3}], but it doesn’t really matter what we call it.) First of all, note that if we take any two numbers in S and add them, we get another number in S:

(a + b\sqrt{3}) + (c + d\sqrt{3}) = (a + c) + (b + d)\sqrt{3}

For example, (2 + 3\sqrt{3}) + (4 - 4\sqrt{3}) = 6 - \sqrt{3}. A shorter way to say this is that S is “closed under addition”. The additive identity, 0, is also a member of S, since 0 = 0 + 0\sqrt{3}.

We can show that S is closed under multiplication, too:

(a + b\sqrt{3})(c + d\sqrt{3}) = ac + bc\sqrt{3} + ad\sqrt{3} + 3bd = (ac + 3bd) + (bc + ad)\sqrt{3}

And the multiplicative identity 1 is likewise in S, since 1 = 1 + 0 \sqrt{3}.

So S is a sort of “system of numbers” in a similar sense to the integers, or rational numbers, or especially the complex numbers a + bi. At this point we could start talking about rings and fields and stuff, but we don’t actually need anything that complicated. We just need to talk a little bit about groups. I’ve been wanting to write about groups on this blog for a while, and this is a great excuse! So, tomorrow, we’ll start to learn about groups and develop some of the ideas we will need to understand \omega.

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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